What is the 'hybrid vortex tube-vorton' method?

The recently developed Full Non-linear Vortex Tube-Vorton Method (FTVM) is a Lagrangian-based unsteady three-dimensional numerical method, that is, it does not require meshes to subdivide the space occupied by the flow, as Eulerian methods do in a more conventional computational fluid dynamics approach, for example, based on finite volumes. For this reason, it is less computationally expensive because, like all particle-based methods, it only requires calculating the flow variables at certain (and sufficient) points of interest and not in the entire surrounding space. In addition, the FTVM is based on a novel circulation-vorticity model, detached from the entire surface, which has recently been peer-reviewed and published in "Advances in Aerodynamics" Journal (Q1), owned by a world-renowned publishing house (Springer). In addition, the corresponding provisional patent was filed at the United States Patent and Trademark Office (USPTO) and recently extended to an international one through the Patent Cooperation Treaty (PCT).

Fig. 1 The Full non-linear Vortex Tube-Vorton Method (FTVM, 2023).

The development of the steady full multi-wake model [1] is based on the vortex lattice method (VLM) and, through a reinterpretation of the Potential Flow Theory, which is the basis of the present unsteady development (FTVM). The first one consisted of an arduous effort of around 8 months of continuous work within the framework of a doctoral project in engineering, which in turn is based on a simplified model developed in the 90s at NASA (code LinAir), which considers the inviscid flow detachment from the surface of a shell-body. Of course, such a multi-wake model complied with the rigor required by such a level of research, following the recommendations and procedures to verify and validate it for different operating conditions, geometries, and alignments to the flow. However, despite the satisfactory results obtained, in the end, it was rejected due to "lack of theoretical support" to present it as an advance that would allow solving for separated flow cases through a vortex method (see a note at the end of the text).

Fig. 2 Circulation detached from the flat plate's surface scheme (code LinAir, 1993; NASA).

Despite the above, the natural development of such research continued independently, giving way to the first unsteady full multi-wake method (UFVLM) [2], which allows simulating inviscid flow past a flat object. Even though in the first instance it may be thought that the results obtained using this particular method contribute little to the current research, which is attributable to the type of vortex element used to discretize the wake (straight elements instead of spherical ones), the algorithm and the level of precision achieved during its verification and validation phases should be considered as the basis for the most recent development, that is, the method based on 'tubes and vortons' [3].

Fig. 3 The Unsteady Full Multi-wake Vortex Lattice Method (UFVLM, 2022).

 To put it in context and be able to make a comparison, the mesh-free vortex particle method (VPM) and its recent reformulation (rVPM), which have been applied to fully attached flow cases (under viscous assumption), mainly in simulations with rotors and propellers, implement additional analytical equations and other numerical schemes to approximate a divergence-free vorticity field, which is a necessary condition to be met to correctly solve an incompressible flow. After having explored and implemented both approaches under different schemes (classical, transposed, and mixed for the calculation of the vortex squeezing/stretching or, simply, vortex stretching), it was found that none of these comply with exactly conserving the total amount of circulation over time (Kelvin's circulation theorem), and their solutions invariably diverge when implemented together with the detached flow model, even for relatively low angles of attack; in fact, this unstable behavior, attributable to the accumulation error in the vortex stretching calculation, also occurs in less complex simulations (from the flow separation viewpoint), according to their own authors; however, according to shown results, the solution tends to stabilize by adding a turbulence model to the rVPM.

Now, regarding the FTVM, which is based on the vortex filament method (VFM), quite similar to the used in the UFVLM, and which offers the great advantage of maintaining both divergence-free velocity and vorticity fields through the interconnection of all wake vorticity elements through a grid along the entire simulation (in absence of vorticity collisions with the shell-body), it is necessary to mention that this interconnection is not enforced but natural, since each tube (isolated regularized filament) moves in space independently as it has its own nodes, so that at each grid's point, there could be up to four overlapping nodes. As far as the calculation of the induced velocity field is concerned, each vortex tube is transformed into its spherical equivalent vorton, which offers the advantage of calculating its influence on its surroundings in a spherical manner but not only along an axis, which is essential to obtaining the results achieved so far. In addition, the volume of each wake's vorton is modified at each iteration to maintain vorticity conservation, thus solving precisely the vortex stretching term through a finite difference scheme, which makes it a perfectly (zero residual) stable method. It also allows for the diffusion of viscosity by increasing the vorton's volume (vortex blob) through the Core Spreading Method (CSM).

Fig. 4 The FTVM with its corresponding divergence-free grid.

Since the FTVM is research in progress (code available: [4]), there is still work to be done to become a serious candidate and compete with current mesh-based software. However, it is being shown that using a Lagrangian vorticity-based approach, it is possible to solve the Navier-Stokes equations in their velocity-vorticity form accurately without the need to incorporate turbulence models to determine the separation of the three-dimensional flow, which are still semi-empirical models that require external parameters based on experiments, a condition that should ideally be avoided to obtain a method as pure as possible (numerically speaking), reducing the input parameters to a minimum and avoiding problems attributable to Eulerian methods, such as numerical dissipation. Even in the best of cases, the direct numerical simulation (DNS), which is considered the most accurate solution achieved by mesh-based methods and which currently remains extremely computationally expensive, could be replaced by its Lagrangian vorticity-based simile in a few years as the current development evolves, which would require fewer computational resources and, for sure, simpler codes.

Fig. 5 Mesh-based direct numerical simulation (DNS). Source: F. Schenk, and R. Vinuesa, "Enhanced large-scale atmospheric flow interaction with ice sheets at high model resolution" (2019). 

Note: A theoretical justification (by independent research) [5] for the surface-generated (and detached) vorticity hypothesis was published after the steady model was made public [1].

[1] Journal article: The Full Multi-wake Vortex Lattice Method: a detached flow model based on Potential Flow Theory | Advances in Aerodynamics | Full Text (springeropen.com)

[2] Pre-print: https://www.researchgate.net/publication/372419147_The_Unsteady_Full_Multi-wake_Vortex_Lattice_Method_a_full_rolled-up_detached_vorticity_approach

[3] Journal article: The Full Non-linear Vortex Tube-Vorton Method: the pre-stall condition | Advances in Aerodynamics | Full Text (springeropen.com)

[4] Open-source code (VS Fortran): CPimentelMx/VortoNeX: The Full Non-linear Vortex Tube-Vorton Method (FTVM) (github.com)

[5] Journal's article: https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/vorticity-generation-and-conservation-on-generalised-interfaces-in-threedimensional-flows/301E4EC11B708F858BC88EE0DC7AA067